Find the point on the hyperbola xy = 8 that is closest to the point (3,0)

label Mathematics
account_circle Unassigned
schedule 1 Day
account_balance_wallet $5

This is in calculus using optimization.

Nov 21st, 2014

The distance between two points in the xy plane is:

D = [(x2-x1)^2 + (y2-y1)^2]^0.5

Let (x1,y1) be the point (3,0), then D is:

[(x2-3)^2 + (y2)^2]^0.5, or to save clutter:

D = [(x-3)^2 + y^2]^0.5

y = 8/x, so D = [(x-3)^2 + 64x^-2]^0.5

dD/dx = 0.5 [(x-3)^2 + 64x^-2]^-0.5 * [2(x-3) - 128x^-3]

D is at the minimum when dD/dx = 0, so solve for:

[2(x-3) - 128x^-3] = 0

x = 4, so y = 2.

The hyperbola is closest to (3, 0) at (4, 2) where the distance is the square root of 5.

Nov 21st, 2014

Studypool's Notebank makes it easy to buy and sell old notes, study guides, reviews, etc.
Click to visit
The Notebank
...
Nov 21st, 2014
...
Nov 21st, 2014
Aug 21st, 2017
check_circle
Mark as Final Answer
check_circle
Unmark as Final Answer
check_circle
Final Answer

Secure Information

Content will be erased after question is completed.

check_circle
Final Answer